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G = C42.189C23order 128 = 27

50th non-split extension by C42 of C23 acting via C23/C2=C22

p-group, metabelian, nilpotent (class 3), monomial

Aliases: C42.189C23, Q8⋊C89C2, D4⋊C817C2, C4⋊C4.25D4, (C2×D4).46D4, C4.4D83C2, C4⋊D8.3C2, C4⋊C8.4C22, (C2×Q8).44D4, C4⋊SD1633C2, C4.54(C4○D8), C4.10D89C2, (C4×C8).17C22, C4⋊Q8.11C22, C4.60(C8⋊C22), (C4×D4).23C22, (C4×Q8).23C22, C2.18(D4⋊D4), C41D4.13C22, C22.155C22≀C2, C2.15(D4.8D4), C22.50C241C2, (C2×C4).946(C2×D4), SmallGroup(128,360)

Series: Derived Chief Lower central Upper central Jennings

C1C42 — C42.189C23
C1C2C22C2×C4C42C4×D4C22.50C24 — C42.189C23
C1C22C42 — C42.189C23
C1C22C42 — C42.189C23
C1C22C22C42 — C42.189C23

Generators and relations for C42.189C23
 G = < a,b,c,d,e | a4=b4=c2=1, d2=b2, e2=a2b2, ab=ba, cac=dad-1=a-1, eae-1=ab2, cbc=dbd-1=ebe-1=b-1, dcd-1=ac, ece-1=bc, de=ed >

Subgroups: 272 in 108 conjugacy classes, 34 normal (32 characteristic)
C1, C2 [×3], C2 [×2], C4 [×4], C4 [×6], C22, C22 [×6], C8 [×3], C2×C4 [×3], C2×C4 [×7], D4 [×8], Q8 [×4], C23 [×2], C42, C42 [×3], C22⋊C4 [×5], C4⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×4], C2×C8 [×3], D8 [×2], SD16 [×2], C22×C4, C2×D4, C2×D4 [×3], C2×Q8, C2×Q8, C4×C8, D4⋊C4 [×4], C4⋊C8 [×2], C42⋊C2, C4×D4, C4×Q8, C4×Q8, C22⋊Q8, C4.4D4, C422C2 [×2], C41D4, C4⋊Q8, C2×D8, C2×SD16, D4⋊C8, Q8⋊C8, C4.10D8, C4⋊D8, C4⋊SD16, C4.4D8, C22.50C24, C42.189C23
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, C2×D4 [×3], C22≀C2, C4○D8 [×2], C8⋊C22 [×2], D4⋊D4 [×2], D4.8D4, C42.189C23

Character table of C42.189C23

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I4J4K4L8A8B8C8D8E8F8G8H
 size 111181622224444488844448888
ρ111111111111111111111111111    trivial
ρ21111-1-111111-1-111-1-111111-1-111    linear of order 2
ρ311111-11111-111-11-1-1-1111111-1-1    linear of order 2
ρ41111-111111-1-1-1-1111-11111-1-1-1-1    linear of order 2
ρ51111-1-11111-1-1-1-1111-1-1-1-1-11111    linear of order 2
ρ61111111111-111-11-1-1-1-1-1-1-1-1-111    linear of order 2
ρ71111-1111111-1-111-1-11-1-1-1-111-1-1    linear of order 2
ρ811111-1111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ9222200-2-2-2-2000022-2000000000    orthogonal lifted from D4
ρ1022220022-2-22002-200-200000000    orthogonal lifted from D4
ρ1122220022-2-2-200-2-200200000000    orthogonal lifted from D4
ρ12222200-2-2-2-200002-22000000000    orthogonal lifted from D4
ρ132222-20-2-2220220-200000000000    orthogonal lifted from D4
ρ14222220-2-2220-2-20-200000000000    orthogonal lifted from D4
ρ1522-2-2002-2002i00-2i0000--2-2--2-2002-2    complex lifted from C4○D8
ρ1622-2-2002-200-2i002i0000-2--2-2--2002-2    complex lifted from C4○D8
ρ172-2-220000-220-2i2i000002-2-22--2-200    complex lifted from C4○D8
ρ182-2-220000-220-2i2i00000-222-2-2--200    complex lifted from C4○D8
ρ1922-2-2002-200-2i002i0000--2-2--2-200-22    complex lifted from C4○D8
ρ202-2-220000-2202i-2i00000-222-2--2-200    complex lifted from C4○D8
ρ212-2-220000-2202i-2i000002-2-22-2--200    complex lifted from C4○D8
ρ2222-2-2002-2002i00-2i0000-2--2-2--200-22    complex lifted from C4○D8
ρ234-4-4400004-40000000000000000    orthogonal lifted from C8⋊C22
ρ2444-4-400-44000000000000000000    orthogonal lifted from C8⋊C22
ρ254-44-4000000000000002i2i-2i-2i0000    complex lifted from D4.8D4
ρ264-44-400000000000000-2i-2i2i2i0000    complex lifted from D4.8D4

Smallest permutation representation of C42.189C23
On 64 points
Generators in S64
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)(25 26 27 28)(29 30 31 32)(33 34 35 36)(37 38 39 40)(41 42 43 44)(45 46 47 48)(49 50 51 52)(53 54 55 56)(57 58 59 60)(61 62 63 64)
(1 18 15 5)(2 19 16 6)(3 20 13 7)(4 17 14 8)(9 62 56 59)(10 63 53 60)(11 64 54 57)(12 61 55 58)(21 31 33 26)(22 32 34 27)(23 29 35 28)(24 30 36 25)(37 44 45 50)(38 41 46 51)(39 42 47 52)(40 43 48 49)
(2 4)(5 18)(6 17)(7 20)(8 19)(9 61)(10 64)(11 63)(12 62)(14 16)(21 26)(22 25)(23 28)(24 27)(29 35)(30 34)(31 33)(32 36)(37 38)(39 40)(41 50)(42 49)(43 52)(44 51)(45 46)(47 48)(53 57)(54 60)(55 59)(56 58)
(1 45 15 37)(2 48 16 40)(3 47 13 39)(4 46 14 38)(5 50 18 44)(6 49 19 43)(7 52 20 42)(8 51 17 41)(9 36 56 24)(10 35 53 23)(11 34 54 22)(12 33 55 21)(25 62 30 59)(26 61 31 58)(27 64 32 57)(28 63 29 60)
(1 28 13 31)(2 30 14 27)(3 26 15 29)(4 32 16 25)(5 23 20 33)(6 36 17 22)(7 21 18 35)(8 34 19 24)(9 51 54 43)(10 42 55 50)(11 49 56 41)(12 44 53 52)(37 60 47 61)(38 64 48 59)(39 58 45 63)(40 62 46 57)

G:=sub<Sym(64)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48)(49,50,51,52)(53,54,55,56)(57,58,59,60)(61,62,63,64), (1,18,15,5)(2,19,16,6)(3,20,13,7)(4,17,14,8)(9,62,56,59)(10,63,53,60)(11,64,54,57)(12,61,55,58)(21,31,33,26)(22,32,34,27)(23,29,35,28)(24,30,36,25)(37,44,45,50)(38,41,46,51)(39,42,47,52)(40,43,48,49), (2,4)(5,18)(6,17)(7,20)(8,19)(9,61)(10,64)(11,63)(12,62)(14,16)(21,26)(22,25)(23,28)(24,27)(29,35)(30,34)(31,33)(32,36)(37,38)(39,40)(41,50)(42,49)(43,52)(44,51)(45,46)(47,48)(53,57)(54,60)(55,59)(56,58), (1,45,15,37)(2,48,16,40)(3,47,13,39)(4,46,14,38)(5,50,18,44)(6,49,19,43)(7,52,20,42)(8,51,17,41)(9,36,56,24)(10,35,53,23)(11,34,54,22)(12,33,55,21)(25,62,30,59)(26,61,31,58)(27,64,32,57)(28,63,29,60), (1,28,13,31)(2,30,14,27)(3,26,15,29)(4,32,16,25)(5,23,20,33)(6,36,17,22)(7,21,18,35)(8,34,19,24)(9,51,54,43)(10,42,55,50)(11,49,56,41)(12,44,53,52)(37,60,47,61)(38,64,48,59)(39,58,45,63)(40,62,46,57)>;

G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48)(49,50,51,52)(53,54,55,56)(57,58,59,60)(61,62,63,64), (1,18,15,5)(2,19,16,6)(3,20,13,7)(4,17,14,8)(9,62,56,59)(10,63,53,60)(11,64,54,57)(12,61,55,58)(21,31,33,26)(22,32,34,27)(23,29,35,28)(24,30,36,25)(37,44,45,50)(38,41,46,51)(39,42,47,52)(40,43,48,49), (2,4)(5,18)(6,17)(7,20)(8,19)(9,61)(10,64)(11,63)(12,62)(14,16)(21,26)(22,25)(23,28)(24,27)(29,35)(30,34)(31,33)(32,36)(37,38)(39,40)(41,50)(42,49)(43,52)(44,51)(45,46)(47,48)(53,57)(54,60)(55,59)(56,58), (1,45,15,37)(2,48,16,40)(3,47,13,39)(4,46,14,38)(5,50,18,44)(6,49,19,43)(7,52,20,42)(8,51,17,41)(9,36,56,24)(10,35,53,23)(11,34,54,22)(12,33,55,21)(25,62,30,59)(26,61,31,58)(27,64,32,57)(28,63,29,60), (1,28,13,31)(2,30,14,27)(3,26,15,29)(4,32,16,25)(5,23,20,33)(6,36,17,22)(7,21,18,35)(8,34,19,24)(9,51,54,43)(10,42,55,50)(11,49,56,41)(12,44,53,52)(37,60,47,61)(38,64,48,59)(39,58,45,63)(40,62,46,57) );

G=PermutationGroup([(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24),(25,26,27,28),(29,30,31,32),(33,34,35,36),(37,38,39,40),(41,42,43,44),(45,46,47,48),(49,50,51,52),(53,54,55,56),(57,58,59,60),(61,62,63,64)], [(1,18,15,5),(2,19,16,6),(3,20,13,7),(4,17,14,8),(9,62,56,59),(10,63,53,60),(11,64,54,57),(12,61,55,58),(21,31,33,26),(22,32,34,27),(23,29,35,28),(24,30,36,25),(37,44,45,50),(38,41,46,51),(39,42,47,52),(40,43,48,49)], [(2,4),(5,18),(6,17),(7,20),(8,19),(9,61),(10,64),(11,63),(12,62),(14,16),(21,26),(22,25),(23,28),(24,27),(29,35),(30,34),(31,33),(32,36),(37,38),(39,40),(41,50),(42,49),(43,52),(44,51),(45,46),(47,48),(53,57),(54,60),(55,59),(56,58)], [(1,45,15,37),(2,48,16,40),(3,47,13,39),(4,46,14,38),(5,50,18,44),(6,49,19,43),(7,52,20,42),(8,51,17,41),(9,36,56,24),(10,35,53,23),(11,34,54,22),(12,33,55,21),(25,62,30,59),(26,61,31,58),(27,64,32,57),(28,63,29,60)], [(1,28,13,31),(2,30,14,27),(3,26,15,29),(4,32,16,25),(5,23,20,33),(6,36,17,22),(7,21,18,35),(8,34,19,24),(9,51,54,43),(10,42,55,50),(11,49,56,41),(12,44,53,52),(37,60,47,61),(38,64,48,59),(39,58,45,63),(40,62,46,57)])

Matrix representation of C42.189C23 in GL4(𝔽17) generated by

0100
16000
00115
00116
,
0100
16000
0010
0001
,
1000
01600
0010
00116
,
5500
51200
00116
00146
,
3300
31400
00130
00013
G:=sub<GL(4,GF(17))| [0,16,0,0,1,0,0,0,0,0,1,1,0,0,15,16],[0,16,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,16,0,0,0,0,1,1,0,0,0,16],[5,5,0,0,5,12,0,0,0,0,11,14,0,0,6,6],[3,3,0,0,3,14,0,0,0,0,13,0,0,0,0,13] >;

C42.189C23 in GAP, Magma, Sage, TeX

C_4^2._{189}C_2^3
% in TeX

G:=Group("C4^2.189C2^3");
// GroupNames label

G:=SmallGroup(128,360);
// by ID

G=gap.SmallGroup(128,360);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,141,456,422,520,1123,570,521,136,2804,1411,718,172]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=b^4=c^2=1,d^2=b^2,e^2=a^2*b^2,a*b=b*a,c*a*c=d*a*d^-1=a^-1,e*a*e^-1=a*b^2,c*b*c=d*b*d^-1=e*b*e^-1=b^-1,d*c*d^-1=a*c,e*c*e^-1=b*c,d*e=e*d>;
// generators/relations

Export

Character table of C42.189C23 in TeX

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